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Estimative and predictive distances. (English) Zbl 0764.62026

Summary: Methods of estimating distances between members of $\left(r,r\right)$ exponential families are considered. The first replaces the parameters in the geodesic distance associated with the information metric by their maximum likelihood estimates. The second is based on the family of predictive densities corresponding to Jeffreys’ invariant prior, using the sufficient statistics as co-ordinates of a Riemannian manifold.

In all examples considered, the resulting estimative and predictive distances differ in form by only a simple multiple, the predictive distance being the shorter, and interesting geometrical relationships associated with flatness are also observed. Finally, the effect of the conjugate priors on distances and flatness is considered.

##### MSC:
 62F10 Point estimation 62F15 Bayesian inference
##### References:
 [1] Aitchison, J. (1975). Goodness of prediction fit.Biometrika 62, 547–554. · Zbl 0339.62018 · doi:10.1093/biomet/62.3.547 [2] Aitchison, J. and Kay, J. W. (1975). Principles, practice and performance in decision-making in clinical medicine.Proc. 1973 NATO Conference on the Role and Effectiveness of Decision Theory. London: English Universities Press. pp. 252–273. [3] Amari, S.-I. (1985).Differential–Geometrical Methods in Statistics, Lecture Notes in Statistics28. Berlin: Springer-Verlag. [4] Atkinson, C. and Mitchell, A. F. S. (1972). Rao’s distance measure. (UnpublishedTech. Rep.) Imperial College, U.K. [5] Atkinson, C. and Mitchell, A. F. S. (1981). Rao’s distance measure.Sankyã A 43, 345–365. [6] Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion).J. Roy. Statist. Soc. B 41, 113–147. [7] Critchley, F., Marriott, P. and Salmon, M. (1992). distances in statistics. Proc.XXXVI Riunione Scientifica, Societa Italiana di Statistica, Rome: CISU, pp. 39–60. [8] Geisser, S. (1966). Predictive discrimination.Multivariate Analysis. (P. R. Krishnaiah, ed.) New York: Academic Press, pp. 149–163. [9] Jeffreys, H. (1948)Theory of Probability, Oxford: University Press. [10] Kullback, S. and Leibler, R. A. (1951). On information and sufficiency.Ann. Math. Statist. 22, 525–540. · Zbl 0042.38403 · doi:10.1214/aoms/1177729694 [11] Mahalanobis, P. C. (1936). On the generalized distance in statistics.Proc. Nat. Inst. Sci. India A 2, 49–55. [12] Mitchell, A. F. S. and Krzanowski, W. (1985). The Mahalanobis distance and elliptic distributions.Biometrika 72, 464–467. · Zbl 0571.62042 · doi:10.1093/biomet/72.2.464 [13] Mitchell, A. F. S. (1988). Statistical manifolds of univariate elliptic distributions.Internat. Statist. Rev. 56, 1–16. · Zbl 0677.62009 · doi:10.2307/1403358 [14] Mitchell, A. F. S. (1989). The information matrix, skewness tensor and $\alpha$-connections for the general multivariate elliptic distribution.Ann. Inst. Statist. Math. 41, 289–304. · Zbl 0691.62049 · doi:10.1007/BF00049397 [15] Murray, G. D. (1977). A note on the estimation of probability density functions.Biometrika 64, 150–152. · Zbl 0347.62035 · doi:10.2307/2335788 [16] Rao, C. R. (1945). Information and the accuracy attainable in the estimation of statistical parameters.Bull. Calcutta Math. Soc. 37, 81–91. [17] Rao, C. R. (1949). On the distance between two populations.Sankyã 9, 246–248.